PSI - Issue 28

Per Stahle et al. / Procedia Structural Integrity 28 (2020) 2065–2071 P. Ståhle et al. / Structural Integrity Procedia 00 (2020) 000–000

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Further, for preservation of volume, while p the thickness direction of the foil, i.e., p p z . At initiation of necking, the thinning, h z , and the hardening rate of the material has reached a balance that prevents the stress from additional increase, meaning that d σ y h (1 + z ) = 0 ⇒ d σ y d z = − σ y 1 + z . (2) The necking initialises at peak stress and is reasonable to assumed to be constant at least close to the propagating tip of the necking region, due to the balance between material hardening and the reduction of thickness, that initialises the necking. The present pilot study is following the development of the neck only until buckling at which the reduction of the cross section assumed to be small. This may seem as a bold assumption but it will be verified in Sect. 4. x = 0, the remaining in-plane strain component, p y , equals the strain in x = −

2.2. The small scale yielding limit

At small scale yielding the stress intensity factor, K I , is related to the crack growth energy release rate G = K 2 I / E , and at small scale yielding

σ 2

√ π a ⇒ G = π

∞ a E

(3)

K I = σ ∞

,

where a is the half length of the crack placed perpendicular to the uniaxial tensile remote stress σ ∞ . Further, the ASTM convention that applies the condition a > 2 . 5( K I c /σ Y ) 2 , cf. Brown and Srawley (2005), where K I c is the fracture toughness, is in the present paper adopted as the limit of small scale yielding and linear fracture mechanics. Using the ASTM condition and Eq. (1) and (3) the following relation between the remote and the cohesive stresses preserving small scale yielding is obtained as,

σ Y √ 2 . 5 π ≈

0 . 31 σ D .

(4)

σ ∞ <

2.3. Failure versus fracture

Metal foils with thicknesses on the µ m scale commonly fails through cross section plastic yielding. At the crack tip the displacement across the necking region, is 2 υ ∗ ≤ h , which at equality initiates crack growth. At small scale yielding this happens if the energy release rate becomes G = σ D h < K 2 I c / E , (5) cf. e.g. Broberg (2005). It leads to plastic failure at a lower load than what is required for conventional fracture and formation of crack surfaces. The following two criteria regarding the foil thickness, is applicable at small scale yielding,

K 2 I c E σ D

h c =

leads to plastic failure if h > h c and fracture if h > h c .

(6)

At large scale yielding the authors believe that fracture may happen at thicknesses smaller than what is stated by above. However, in the present paper it is assumed that the foils are su ffi ciently thin for the di ff erent cases studied. Taking aluminium as an example, the foil thicknesses leading to fracture instead of failure are around a millimetre while a consumer household rather soft type of aluminium foil, and normally thinner than 20 µ m.

2.4. Buckling analysis

For short cracks, the buckling stress, is given by ξ E ( h / a ) 2 , where ξ is a function of Poisson’s ratio, ν , and the ratio of remote stress and yield stress, σ ∞ /σ Y ., cf. Marksto¨m and Storåkers (1980), Brighenti (2009). The buckling analyses rely on stress distributions calculated under the assumption of absent anti-plane displacements, w .